Hydromagnetic Equilibria and Force-Free Fields

Transcription

1 P/386 USA Hydromagnetic Equilibria and Force-Free Fields By H. Grad and H. Rubin* INTRODUCTION ELEMENTARY PROPERTIES The equations governing the equilibrium of a perfectly conducting fluidin the presence of a magnetic field are Vp = JxB V.B -О or V{p + B 2 \2 x) = (B-V)B/^ V.B = 0, (1) p is the fluid pressure, В the magnetic field, and J the current density. These equations admit a large variety of solutions, i.e., of equilibrium configurations in which a conducting fluid is balanced by a magnetic field. It is our purpose to survey these possibilities with the expectation that, if a solution has been found to exist mathematically (and is, in addition, stable) it can actually be constructed by sufficient exercise of experimental ingenuity. In this context, it is extremely important to discover exactly what data should be specified in order to determine a solution uniquely. In addition to certain general properties of these equations, we shall consider their solution in terms of arbitrary functions, the solution of well-posed boundary value problems, and several alternative formulations of the problem in terms of the calculus of variations. In principle, either the differential equations or the variational characterization can be taken as the definition of equilibrium the two are only approximately equivalent. Moreover, a certain type of variational formulation (slightly different from that treated in this paper) can be instrumental in a stability analysis, which is essential to give physical meaning to an equilibrium configuration. 1 By inspection of (1), since В and J are perpendicular to V, p, we see that p is constant on В lines and on J lines equivalently, the В lines and J lines lie on constant p surfaces. In the case of a unidirectional magnetic field (e.g., B z (x,y)) the general solution is p + B 2 /2 u = constant. The fluid pressure is balanced by the * Institute of Mathematical Sciences, New York University. magnetic pressure either p or В can be given arbitrarily and the other one " filled in ". The integral form of (1) is = 0, These equations are more fundamental than the differential equations (1). They are equivalent to (1) when the functions are smooth. In addition, at a discontinuity surface, the integral relations (2) imply that B n vanishes and that p + B 2 \2 x is continuous. A discontinuity surface must be a flux surface otherwise the Maxwell magnetic stress tensor would not be compatible with a scalar pressure. A FLUID-DYNAMICAL ANALOGUE Using an appropriate identification of variables, equations (1) become identical to the equations of steady incompressible inviscid flow. Setting the fluid density equal to unity, these equations are (u.v)u + VP* V.u -0 p* denotes the fluid pressure. The identifications are u ~ B/V/ and p* ~ p + B 2 \2 x the negative of the pressure p in the magnetic case then corresponds to the Bernoulli constant p* + u 2 /2. The interesting case is rotationalflow,for which the vorticity Vxu^O corresponding to J Ф 0. It is interesting to consider the analogue of the fluid-dynamical free boundary or cavitation problem in which an irrotational flow is separated at a discontinuity surface (vortex sheet) from stagnant fluid or a cavity. The separation surface is determined by the extra boundary condition u = constant. This is mathematically equivalent to a vacuum magnetic field (J = 0) separated at an interface (current sheet) from a field-free conducting fluid. Since p is constant in the conducting fluid, we obtain the free boundary condition B constant. Although the two problems are mathematically identical, it is perfectly possible for a significant fluiddynamical problem to be uninteresting in the magnetic analogue (e.g., a jet from an orifice, Fig. la) and vice versa (e.g., the cusped equilibrium, 2 Fig. lb). 0 (2) (3) 190

2 HYDROMAGNETIC EQUILIBRIA AND FORCE-FREE FIELDS 191 SOLUTION OF BOUNDARY VALUE PROBLEMS The characteristics of a system of partial differential equations 3 give immediate qualitative and possibly quantitative information concerning relevant initial value or boundary value problems. An elementary calculation yields four characteristics for the system (1). Two of the characteristics are purely imaginary as for the potential equation. Corresponding to these two characteristics, one would expect to prescribe a single scalar boundary value on the entire boundary of the domain, e.g., the normal component B n of the magnetic field. The remaining two characteristics are real viz., the В lines counted twice. Corresponding to each real characteristic, one would expect to be able to specify a single scalar quantity at one end of each В line. In a geometry as in Fig. 2 in which every В line Figure la Figure 1b intersects each end of the tube, one would expect to give both additional quantities at one end or one at each end. For example, one may conjecture that the following specification of boundary values (in addition to B n on the whole surface) would be appropriate t, n designate tangential and normal components : Problem A x : p given on S lf p given on S 2 (compatibly) Problem A 2 : p given on S v J n given on S 1 Problem A 3 : p given on S lt J n given on 5 2 Problem A 4 : B t given on S x (as well as B n : the vector В is thus given). In these problems, we prescribe p on boundary surfaces in such a way that the p lines are " simple " (not closed) as shown in Fig. 3. Moreover, in Problem A lf since p is constant on В lines, matching magnetic flux on both ends imposes a simple compatibility requirement on p. For example, if p is given on S x and the lines of constant p are specified on S 2, this condition fixes the values of p on these given lines. An iteration scheme for Problems A 2 and A 3 offering likelihood of convergence is as follows: Figure 2 Figure 3 suppose that, at a certain stage of the iterations, we have a vector field В satisfying V»B = 0 and the boundary condition for Ъ п - We find p every in the domain by carrying the boundary values of p along these В lines. The component J ± perpendicular to В is then obtained in the domain from the equation V.p = JxB. We write J,, = ob for the parallel component and employ the requirement V»J = 0 to obtain along each В line the ordinary differential equation B.Vcx + V-J ± = 0 (4) for a. The given " initial " condition for J n at one end allows this equation to be solved uniquely on each line so that J is determined every. We now solve for a new В from V-B = 0, V X В = J and the boundary condition for B n, and then we continue the iterations. We next consider two-dimensional problems i.e., problems in which no quantity depends on z. There are several possibilities: (1) В can have the single component B z and J the two components J x and ] y \ (2) В can have the two components B x and B y and J the single component / z (3) both В and J can be general (three-component) vectors depending on x and у alone. The general solution in the firstcase has already been given explicitly, p + B 2 2[Á constant. In the second case, the number of characteristics is three rather than four the В lines are counted only once. The two-dimensional analogue to Problem A x will be considered later. Corresponding to Problems A 2, A 3 and A 4 we have (see Fig. 4) Figure 4 Problem B x : p given on С г or C 2 Problem B 2 : В given on С г or C 2. In the third (general two-dimensional) case, the characteristics are counted as in the full three-dimensional case. Correspondingly we list: Problem C 2 : p given on Y lt J n given on I\ Problem C 3 : p given on I\, J n given on Г 2 Problem C 4 : В given on I\ (as above, the analogue to Problem А г is left to later). The two-dimensional Problems C 2 and C 3 are equivalent. With axial symmetry, the в coordinate corresponds to the z coordinate of the two-dimensional case. We have the same three subdivisions: (1) B e, J r, ] z \ (2) B r> B Zy J e \ (3) В and J general (three-component) vectors depending on r and z alone. In the first case, B e and p must be functions of r alone such that

3 192 SESSION A-5 P/386 H. GRAD and H. RUBIN d 8r(p (5) In the second and third cases, we identify Problems D-L, D 2 and E 2, E 3, E 4 in correspondence with B v B 2, and C 2, C 3, C 4, respectively. Additional justification of the above conjectures will be given later here they are suggested merely by the counting of characteristics. The problem of force-free fields, viz., solution of the system (VxB)xB = 0 V.B = 0 (6) is a special case of the equilibrium problem obtained by specializing boundary values of pressure to p = constant. We identify Problem F 2 as the special case of A 2 or A 3, F 2 as the special case of C 2 or C 3, and F 3 as the special case of E 2 or E 3. Problems F 2 and F 3 are the two-dimensional and axially symmetric versions of г. CHARACTERIZATION OF THE MAGNETIC FIELD IN A PRESSURE SURFACE We first introduce the concept of a surface harmonic vector field on a surface S (e.g., a constant p surface). A tangential vector field X( 2 ) = V (2)^ is said to be a surface gradient (or irrotational vector field) if it is the projection of a three-dimensional gradient or if = 0 for every closed curve С which bounds a portion of 5. The conjugate vector field nxx^2) = Y< 2 ) ( n is the normal to 5) is said to be a surface curl (or solenoidal vector field) we have here ф and гр are conjugate harmonic with respect to a positive weight function a. Since J has no component normal to a pressure surface S p, we conclude that В is a surface gradient on S p ] i.e., В = Щ- п(дф/дп). (8) In Appendix I, it is shown that, since V«B = 0 and p is constant on В lines, there exists a function w such that В = Vp X Voo = \Vp\n X V( 2 )co. (9) We see that В is a weighted surface harmonic with weight \Vp\ on any S p. The weight \Vp\ arises in converting from an actual three-dimensional solenoidal vector with V*B = 0 to a two-dimensional area-weighted solenoidal vector. We note that В will be an actual three-dimensional gradient instead of a surface gradient only if J = 0 and, hence, p is constant. These remarks allow us to extend the previously conjectured existence theorems to cases in which the p lines on the boundary are closed curves (see Fig. 6). On a surface S p, the magnetic field is determined only when, in addition to the normal component of В at each end, we give a period, e.g., the mmf = Ф В-йс (10) on a curve С circling S p. This argument suggests the following modification of Problem A x : Problem G x : p given on S lt p given on S lf т given for each p. X <2).dx Y v Wds = 0 v denotes the normal in S to the curve C. If X( 2 )=V ( V = nxv ( > (that is, if X< 2 ) is both irrotational and solenoidal), we call it harmonic and say that ф and гр are conjugate surface harmonics. In the special case S is a plane, ф and гр satisfy the Cauchy-Riemann equations. In a simply-connected plane domain, a harmonic vector X< 2 ) is uniquely determined by the boundary values of the normal or tangential components of X( 2 ). Specifying either at the boundary is equivalent (except for a trivial additive constant) to specifying гр or ф at the boundary. Exactly the same facts hold on an arbitrary surface 5. In a multiply-connected domain, the solution to such a boundary-value problem is uniquely determined only when certain additional data called periods are prescribed. These may be specified values of the circulations <fx( 2 Wx on each independent closed circuit or of the fluxes X v Wds on arcs which cut these circuits (see Fig. 5). If these periods are nonzero, the functions ф and гр are multiple-valued. The above theory can be generalized to include weighted surface harmonics which satisfy the equation X(2) = (7) Figure 5 We are now also able to insert the two-dimensional (G 2 ) and axially symmetric (G 3 ) analogues of A x : Problem G 2 : p given on I\, т given for each p\ Problem G 3 : p given on S lt r given for each p. In Problem G 2, т is defined as the line integral of В over a finite distance z, interpreting the figure to be periodic in the z direction. In Problem G 3, r = 2nB e r. In general, the value of r(p) can be interpreted as the degree of " twist " of the magnetic lines on each tubular p surface. It should be noted that J n is an alternative way of specifying this twist. SOLUTION IN TERMS OF ARBITRARY FUNCTIONS First we consider the general two-dimensional case in which B x, By, B z depend only on x, y. Since

4 HYDROMAGNETIC EQUILIBRIA AND FORCE-FREE FIELDS 193 and (20) Figure 6 Figure 7 V»B = 0, we can introduce a stream function гр(х, у) for the B x, B y components of the field we then have дгр дгр (11) so that ду у дх B=nxV^+n5 2 (12) n=(0, 0, 1). It can be seen that p and B z are constant on the curves of constant гр in the x,y plane. However, p and B z may be multiple-valued functions of гр. The stream function гр(х,у) satisfies the non-linear potential equation /0=0 (13) (14) p' and /' refer to derivatives with respect to гр. In (13), р(гр) and (гр) are arbitrary functions of гр. Hence, our system of four first-order equations has been reduced to a single second-order equation containing two arbitrary functions. One can expect a solution to be determined by specifying the arbitrary functions and boundary values for гр as for the standard two-dimensional potential equation. For solutions depending on x only, the equation (15) can be integrated explicitly giving many interesting equilibrium configurations. In the case of axial symmetry in which B r, B e, B z depend only on r, z, we introduce a stream function ip(r, z) such that and, hence r Tz 1 дгр (16) In the case of cylindrical symmetry, in which there is dependence on r alone, we may give any two of p, гр and B e as functions of r and immediately compute the other. Interesting special solutions of Eq. (15)andEq. (18) can be found by separation of variables after taking /' and p' to be linear in гр. In some cases eigenvalue problems arise. It is thus clear that uniqueness cannot be expected in general. However, it is possible to prove uniqueness as well as existence even with quite arbitrary (гр) and р(гр) for domains which are not too large. The force-free special cases are obtained by the simple expedient of setting p' (гр) 0. 4 The reduction in terms of arbitrary functions offers support for the conjectured existence of solutions to Problems ~B V C 2 or C 3, D lf E 2 or E 3, F 2, and F 3. In each case, the boundary data over and above B n serve to determine the functions р(гр) and (гр). VARIATIONAL ANALYSIS. ADDITIONAL BOUNDARY VALUE PROBLEMS It is well known that solutions of the fluid freesurface problem can be described as those vector functions n(x) which make stationary the functional \\u 2 dv when the fluid volume is held fixed. Analogously, in the magnetic case we vary \B 2 \2 idv holding the volume of the magnetic domain fixed. Or we can drop the restriction to constant volume and, as suggested by the Lagrange multiplier rule, vary J Vm J Vm 2/л I PodV-{- constant = I (- 7 B 2 p)dv+constant. Jv f+ v m 2V Here V m represents the vacuum (magnetic) domain and Vf the conducting fluid domain the sum V m + Vf is fixed. The pressure p takes the value zero in the vacuum and the constant value p 0 in the fluid. We shall see that, if interpreted properly, the same functional, JVf В + nb d (17) r i (22) n = (n r, n 0, n z ) = (0, 1, 0). Here гр (r, z) satisfies the equation =0 (18) (19) serves for the general equilibrium problem with fluid and magnetic field mixed here p as well as В will be a function of x, y, z. The class of admissible pairs (B, p) allowed to compete in the variation of the given functional is restricted by the conditions that В be solenoidal and p be constant on the В lines of the associated B,

5 194 SESSION A-5 P/386 H. GRAD and H. RUBIN (23) as well as by boundary conditions that will vary from problem to problem. A simple way of incorporating both constraints (23) is to represent В in the form В (24) (see Appendix I). The function со may be multiplevalued if the p surfaces are not simple. We perform the variation, obtaining = j v {B. {VôpxVco+VpxVôœ) - ôp}dv = { v {ôp(j-vœ-l)-dœ(j.vp)}dv -(j) {dp{bxvco)-dœ{bxvp)}*nds = 0. (25) For arbitrary variations of dp and ôco in the volume integral, we conclude J. Vco = 1 J-Vp =0 (26) from which we easily obtain \Jp = J x В as the Euler equation. Next we turn to the boundary variation and first consider the case of simple p surfaces (Fig. 3), with p given at both ends Si. It can easily be verified that there is no contribution to the variation (25) from S 3 on which B n = 0. On 5j, from ôb n = 0we conclude that ôco is a function of p however, we are certainly free to fix the value of со at one end of each p line on S 1} and this makes coô = 0 every. We then have : Theorem 1 : In the tubular geometry of Fig. 2, if p is given (compatibly) at the ends S ± in such a way that the p lines are simple, then Q is made stationary by any solution of the system (1) which satisfies these boundary conditions. This theorem corresponds to the conditions of Problem A v Next consider tubular p surfaces (Fig. 6). Since со and ф (see Eq. (8)) may be multiple-valued, we cut the domain on a surface S which extends from the axis of the nested surfaces S p to the outer boundary S 3 (Fig. 7), The boundary integral in (25) must now be taken over both sides of S as well as S lf S 2 and S 3. As in the previous case, the contribution from S 3 vanishes. A little manipulation shows that the contribution from the cut S also vanishes. On S x or S 2, since ôb n = 0, we must have а со = f(p). We obtain for this part of the variation, S x to S 2 on a given pressure surface. In order to have Q stationary we must either fix this twist beforehand so that ôco 2 ~ ôco 1 = 0 or else specify that r(p) == 0 the latter is a natural boundary condition in this problem. To specify the twist for a class of admissible vector fields B, we require the magnetic lines which originate on a ray С on Б г to end on another given ray C" on S 2 (Fig. 8). We have: Theorem 2: With tubular p surfaces, p given compatibly at the ends S ± and two given rays С and C" identified, Q is stationary if В and p satisfy (1). If the twist is not specified, Q is stationary for a solution of (1) which satisfies r(p) = 0. There are really two ways of specifying twist, since the specification of r(p) or the identification of two rays С and C" are, in an intuitive sense, equivalent. It is easy to alter the problem by the use of the Lagrange multiplier rule so as to be able to specify r instead of the twist. This variational formulation corresponds to Problems G lf G 2 and G 3. Some modifications of these variational problems are necessary to obtain force-free fields. We drop the term p in the variational functional, and obtain M= í B4V. 2/л Instead of the representation B=VpXVoj take (28) we now n is no longer identified with the pressure. is now easy to verify the following theorems: (29) Theorem 3: For admissible sets (B, n) with simple TI surfaces and n given compatibly on S 1, M is stationary when В is force-free. Theorem 4: For admissible sets (В,л) with tubular n surfaces and n given compatibly on S x as well as the twist from S x to S 2, M is stationary when В is force-free. If one is willing to place a large burden on the verification of the compatibility of given boundary data, then it is possible to formulate problems in which p (or n) is given at both ends S x without regard to the simplicity of p lines. We have: Theorem 5 : Consider admissible classes {B,p) [or (В, л)] which satisfy V «B = 0 and p [or л] constant on В lines, p [or jr] given compatibly at both ends, and a fixed (compatible) identification of the ends of each В line. Then Q [or M] is rendered stationary when (B, p) satisfies V = JxB [JxB = 0]. It (27) using Eq. (11.11) of Appendix II. The difference дсо 2 dco x represents the variation of the twist from Figure 8

6 HYDROMAGNETIC EQUILIBRIA AND FORCE-FREE FIELDS 195 This theorem follows when we note that assignment of B n and p [or 7t\ at both ends uniquely fixes the correspondence between В lines if the p surface \_n surfaces] are simple, and if they are not simple, the additional specification of twist serves the same purpose. Physically, the boundary condition that the ends of each В line be fixed corresponds to a perfectly conducting fluid in contact with perfectly conducting end walls S x. In the force-free case, the reference is to a so-called " pressureless plasma " in which the gas pressure is negligible compared to the magnetic pressure, but which is, nevertheless a good conductor that is physically realizable since the conductivity of a plasma is independent of its density. It is interesting to compare the variational problem for the force-free field with the classical Dirichlet's principle which states that M is minimized subject to given B n (and V-B = 0) when VxB = 0. The additional requirement that the ends of each В line be fixed prevents this minimum from being attained and yields (VxB)xB = 0 as the Euler equation instead of VxB = 0. As a final example, we take as our domain the interior of a topological torus and look for solutions which have the outer boundary of the torus as a constant pressure surface. No boundary values can be given since no В lines are accessible. However, the variational formulation itself suggests what data are required to obtain a well-posed problem. As before, we take admissible sets (B, p) which satisfy В = VpxVœ, the p surfaces now are nested toruses about some closed curve C o as axis. For simplicity we take p to be monotone, p 0 > p > p lf p 0 is the value taken on the axis C o and p x is the value taken on the outer boundary. To make со single valued, it is necessary to introduce two surfaces S x as cuts S x is a transverse cut across the torus (leaving ends similar to Б г of Fig. 2), extends from the axis C o to the outer boundary. The only contribution to the variation (25) is on the cuts we have ÔQ = [ [ôp~]{vcoxb).nds (30) - Г [ôœ]vpxb)-nds. Since p (therefore ôp) is single valued, [dp] = 0. The periods [co] are given by Ci (31) C are the independent closed curves on the torus S v. In Appendix II it is shown that (32) Фг(а) represents the magnetic flux across that part of Si for which р г < p < a. From Eq. (11.11) of Appendix II, we obtain the form (33) p 0 and р г are the values taken by p on the axis C o and on the outer boundary of the torus, respectively. We now state: Theorem 6 : In the class of admissible (B, p) defined by V.B = 0, B*Vp = 0 and given Ф г (р), Ф 2 {р) (or а г (р), o 2 (p)) for p o >p>p!, Q is stationary when Vp=JxB. 5 We can obtain a formulation with given ri(p) rather than Фг(р) by modifying Q according to the conventional Lagrange multiplier rule. For force-free fields, exactly the same analysis yields the result that Q is stationary if Ф г is a given function of Ф 2 and Ф 2 attains the fixed values Ф 2 and 0 on C o and on the outer boundary in parametric form we give Ф г (п) and Ф 2 {п). As a special case of a toroidal geometry, we can take a problem with cylindrical symmetry (dependence on r alone) and introduce periodicity to provide the toroidal topology. In this case, the above conjectures are trivially proved. We summarize various sets of data which are believed to define definite equilibrium configurations. In the tubular domain of Fig. 2, we specify B n as shown and we also specify p and J n at one end or p at one end and J n at the other. The justification lies in the counting of characteristics, in a heuristically appealing iteration scheme and in a direct verification by integrating in terms of arbitrary functions in certain special cases. Or we can specify p at one end and the corresponding terminal points of each В line in a manner compatible with the given values of B n. This is confirmed by counting of characteristics supplemented by the known properties of surface harmonics, by variational analysis and by integration in terms of arbitrary functions in certain special cases. In the toroidal geometry, we conclude that the two fluxes Ф%{р) or mmfs r%(p) can be specified arbitrarily the justification is proved by variational analysis and by explicit solution in special cases. APPENDIX I REPRESENTATION OF AN INCOMPRESSIBLE VECTOR FIELD IN TERMS OF TWO STREAM FUNCTIONS We note, for arbitrary <f>(x,y,z), ip{x,y,z) and arbitrary ос(ф, ip) (which means that oc is constant on the intersection of ф = constant and ip = constant), that V-(aV^xV^) 0. (I.I) We remark further that, if (I.I) holds for o {x ) y,z) J ф(х,у, z), y)(x,y, z) in a domain ф and ip are independent functions (i.e., V^xV^^O), then it follows that <x = <х(ф, y>).

7 196 SESSION A-5 P/386 H. GRAD and H. RUBIN Next consider a solenoidal vector field B and a small region of space which is simply covered by the В lines which intersect a transverse surface S. We have: Theorem I.I: There exist functions ф, гр, а(ф, гр) such that В = осх7фхх7гр. To prove Theorem LI we choose any independent ф and гр on S (i.e., ф = constant and гр = constant form a coordinate system). We then carry the values of ф and гр off S on the В lines. Since Х?ф X V^ is parallel to B, we conclude that В = осуфхчгр and, hence, OL = ос(ф, гр) by the above remark. We note that the flux through S, == B n ds = афагр, (1.2) which justifies the terminology " stream functions ". We again take an arbitrary vector field В with V В = 0 and a small region and assert : Theorem 2 : There exist ф, гр such that В = V^ xvf In fact, if ф is any given function which is constant on В lines, then there exists гр such that В = ЧфхУгр. Choose ф arbitrarily on S so that the ф curves are simple. Introduce s as the arc length on а ф curve, v as the normal to the ф curve in S. Note that if В = ЧфхЧгр then B n = (дф/дгцдф/дб). This suggests that we construct гр as f(s)=\~^ds, (1.3) J B integrated along each ф line (the value of гр at one end of each ф line can be arbitrarily assigned). As before, we carry the values of ф and гр off S as constants on В lines. From the previous theorem, В = с фхх7гр. By the construction of гр } a = 1 on S since a is constant on each В line, a = 1 throughout. If the ф lines are taken as closed curves, then the construction yields а гр which is not single valued, and a cut should be introduced. APPENDIX II INTEGRATION FORMULAS We shall use two basic formulas. First, given two functions f(x) and g(x) defined in a space x (%>..., x n ), we have here S is an arbitrary surface in three-space with С as its boundary. 7 The surface S may have to be cut to make ф and гр single valued. If S is a torus, on which there are the two independent closed curves С г and C 2, this formula reduces to f (LIA) s are the periods of ф and гр, respectively. We apply these formulas to obtain the identity for an equilibrium B, i.e., for В satisfying В = У ф == VpxVœ. Using (II.2) and then (II.3) we verify (B 2 ~} v I Js 9 JSt ~Js P ds \vp\ С 2 Г C0X ~J In the special case of a torus, we have j B4V = I {г г {р)а 2 {р) r 2 {p)i Ja<b<p J Oi(p) = n(p) с, B-dx. On a transverse surface S, we compute (И.7) (11.8) (II.9) (11.10) 1<Р<ь (BxV/>.n) S = -j\(p)dp. (11.11) We have n as the normal to S, and introduce v as the normal in S to a p curve, and S as arc length along a p curve (Fig. 9). The proof is = [ f{ds/\vg\). (II.2) The integral (II. 1) over the shell a < g < b is written as an iterated integral first on the surface S g, then with respect to g. 6 The second identity is, J s =J S афагр =J c фагр = ~j c граф (II.3) Figure 9

8 HYDROMAGNETIC EQUILIBRIA AND FORCE-FREE FIELDS 197 <В х V^>.n>v= Í (В x V^.n) Jcp = 1 (BxV.n)^S = ф B*dx = - s Cp *J Cp Again, on a transverse surface S, <B n ) p = f [Vp X Vco-n) Jc p Consequently, or ds ds = o(j>). (11.13) (11.14) (т(/.) = Ф'(#). (11.15) Taking signs into account, we can rewrite (II.9) as L P < b B4v=f b ( т 1 ф 1 # +*«*«REFERENCES 1 J Berkowitz, H Grad and H Rubin, Problems гп Magnetohydrodynamic Stability, P/376 this Volume, these Proceedings, I В Bernstein, E A Fneman, M D Kruskal and R M Kulsrud, An Energy Principle for Hydromagnetic Stability Problems, U S Atomic Energy Comm, Report NYO 7315, Matterhorn Project, Princeton University, Princeton, N J 2 J Berkowitz, К О Friednchs, H Goertzel, H Grad, J Killeen and E Rubin, Cusped Geometries, P/1538, this Volume, these Proceedings 3 R Courant and D Hubert, Methods of Mathematical Physics, First English edition, Interscience, New York- London, Volume II, to appear (in German, Springer, Berlin, 1937) 4 H Furth, M A Levme and R W Wamck, Production and Use of High Transient Magnetic Fields II, Rev Sci. Instr, 28, No 11, (1957) 5 Results similar to these have also been obtained by M D Kruskal Private communication 6 A I Khmchm, Mathematical Foundations of Statistical Mechanics, p 34, Dover, New York 7. A A Blank, H Grad and К О Friednchs, Theory of Maxwell's Equations Without Displacement Current, New York University, Institute of Mathematical Sciences (1957) Report No NYO-6486